Problem

(a) Use de Moivre’s theorem to show that

$$\begin{align*} \sec 5\theta =&\, \frac{\sec^5\theta}{5\sec^4\theta - 20\sec^2\theta + 16}. \end{align*}$$

(b) Hence, obtain the roots of the equation

$$3x^5 - 10x^4 + 40x^2 - 32 = 0$$

in the form $\sec(q\pi)$, where $q$ is rational.